f(x)= 1x has domain and range R 0, asymptotes x=0, y=0, and f-1=f. For f(x)= ax+bcx+d, c 0 and ad-bc 0, the vertical asymptote is x=- dc and horizontal asymptote is y= ac. Find intercepts by setting the numerator or x to zero, provided the original denominator is nonzero. Example 2x+3x+1=2+ 1x+1: asymptotes x=-1, y=2; intercepts (- 32,0) and (0,3). Check cancellation before asserting a vertical asymptote. 2x+2x+1=2, x!= -1, has a missing point at (-1,2), not a vertical asymptote.

\(\mathrm{f}(x)=\dfrac1x\) has domain and range \(\mathbb R\setminus\{0\}\), asymptotes \(x=0\), \(y=0\), and \(\mathrm{f}^{-1}=\mathrm{f}\).

For \(\mathrm{f}(x)=\dfrac{ax+b}{cx+d}\), \(c\ne0\) and \(ad-bc\ne0\), the vertical asymptote is \(x=-\dfrac dc\) and horizontal asymptote is \(y=\dfrac ac\). Find intercepts by setting the numerator or \(x\) to zero, provided the original denominator is nonzero.

\(\dfrac{2x+3}{x+1}=2+\dfrac1{x+1}\): asymptotes \(x=-1\), \(y=2\); intercepts \(\left(-\dfrac32,0\right)\) and \((0,3)\).

Check cancellation before asserting a vertical asymptote. \(\dfrac{2x+2}{x+1}=2\), \(x\ne-1\), has a missing point at \((-1,2)\), not a vertical asymptote.

Graphing Techniques

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Reciprocal and linear rational functions

Reciprocal and linear rational functions